In practically every episode, there is reference to our, now dead, civilization in thebackgrounds. Broken buildings and cars and the like are usually partially buried in a lot of settings. In the opening of the show, dud nuclear missiles and broken pieces oftech are scattered in a wasteland and a gray and red (zombie-like) arm reaches out of a tree.
The earth is shown to have a chunk of it missing (It looks like North/SouthAmerica is almost gone?). In the episode “The Linch”, the group travels over a body of water following the Linch to his lair, where he appears to use toxic waste (possibly fromthe nuclear war) to fuel his power. His hideout is in a destroyed subway station, and upon entering Finn and Jake are attacked by possesed modern-dressed skeletons, as if thepeople died in a bombing waiting for their train. On the surface just outside, tanks and more dud bombs can be seen.
In the episode “Susan Strong”, the Hyoomen species livesin an underground abandoned city, where humans may have tried to live after/during the war before dying off or mutating.
It is mentioned several times in the series thatFinn is the last human, or that humans haven’t been seen for a long time. Marceline and the Ice King are speculated to be survivors of the the Mushroom War because of their ages.